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Existence of stationary solutions for some systems of integro-differential equations with superdiffusion. (English) Zbl 1376.35007

The authors establish the existence of solutions of a system of integro-differential equations arising in population dynamics in the case of anomalous diffusion.
The key ingredients of the proof are a fixed point technique and the use of solvability conditions for elliptic operators without the Fredholm property in unbounded domains.
More specifically, the authors analyze existence of stationary solutions of the following system of integro-differential equations, which appears in cell population dynamics: \[ \frac{\partial u_s}{\partial t} = -D_s\sqrt{-\Delta}u_s +\int_{\mathbb{R}^d} K_s(x-y)g_s(u(y,t))dy+f_s(x), 1\leq s\leq N. \] The space variable \(x\) corresponds to the cell genotype and \(u_s(x,t)\) are densities for different groups of cells. The right-hand side of this system describes the evolution of cell densities by means of cell proliferation, mutations and cell influx. The anomalous diffusion term corresponds to the change of genotype via small mutations and the nonlocal terms describe large mutations. The functions \(g_s\) are the rates of cell birth and \(K_s(x-y)\) describes the proportion of newly born cells changing their genotype from \(y\) to \(x\).
Setting \(D_s=1\), the authors study, under certain conditions, the existence of solutions of the system \[ -\sqrt{-\Delta}u_s+\int_{\mathbb{R}^d} K_s(x-y)g_s(u(y))dy+f_s(x)=0, \;1\leq s\leq N. \] They consider the case where the linear part of the operator above does not satisfy the Fredholm property, and use the method of contraction mappings and solvability conditions for non Fredholm operators to obtain their result.

MSC:

35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
92D25 Population dynamics (general)
47F05 General theory of partial differential operators
35Q92 PDEs in connection with biology, chemistry and other natural sciences
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